An integral with exponential, and trig functions within trig functions

[b3n5p34km4n]

I'm working with the integral from 0 to infinity of

t^(x-1)e^(-atcos(b))cos(atsin(b))

with respect to t. specifically, I'm asked to solve in terms of the gamma function. my question is more of what general technique i should use. all I've been able to do so far is beat it to death using integration by parts, with several different choices for the parts. the main problem i see here is the trig function within the trig function. any insights?

thanks in advance.

 

[SteamKing]

Unless b is a function of t, the integrand is t^(x-1) * e^(-A1*t) * cos(A2*t), where A1 = a*cos(b) and A2 = a*sin(b), and both A1 and A2 are constants.

 

[pwsnafu]

I'm working with the integral from 0 to infinity of

t^(x-1)e^(-atcos(b))cos(atsin(b))

with respect to t. specifically, I'm asked to solve in terms of the gamma function. my question is more of what general technique i should use. all I've been able to do so far is beat it to death using integration by parts, with several different choices for the parts. the main problem i see here is the trig function within the trig function. any insights?

thanks in advance.

You are integrating with respect to t, so the sin(b) and cos(b) are constants. The trig-inside-a-trig is illusionary.